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The Physics-Informed Neural Network (PINN) approach is a new and promisingway to solve partial differential equations using deep learning. The $L^2$Physics-Informed Loss is the de-facto standard in training Physics-InformedNeural Networks. In this paper, we challenge this common practice byinvestigating the relationship between the loss function and the approximationquality of the learned solution. In particular, we leverage the concept ofstability in the literature of partial differential equation to study theasymptotic behavior of the learned solution as the loss approaches zero. Withthis concept, we study an important class of high-dimensional non-linear PDEsin optimal control, the Hamilton-Jacobi-Bellman(HJB) Equation, and prove thatfor general $L^p$ Physics-Informed Loss, a wide class of HJB equation is stableonly if $p$ is sufficiently large. Therefore, the commonly used $L^2$ loss isnot suitable for training PINN on those equations, while $L^{\infty}$ loss is abetter choice. Based on the theoretical insight, we develop a novel PINNtraining algorithm to minimize the $L^{\infty}$ loss for HJB equations which isin a similar spirit to adversarial training. The effectiveness of the proposedalgorithm is empirically demonstrated through experiments. Our code is releasedat https://github.com/LithiumDA/L_inf-PINN.

Is $L^2$ Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?